# New Perturbation Bounds for the Unitary Polar Factor

### Ren-Cang Li

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-94-852

December 1994

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-852.pdf

Let
*A* be an
*m* x
*n* (
*m* >=
*n*) complex matrix. It is known that there is a unique polar decomposition
*A* =
*QH*, where
*Q**
*Q* =
*I*, the
*n* x
*n* identity matrix, and
*H* is positive definite, provided
*A* has full column rank. This note addresses the following question: how much may
*Q* change if
*A* is perturbed? For the square case
*m* =
*n* our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's (
*SIAM J. Matrix Anal. Appl., 14 (1993), 588-597.*). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.

BibTeX citation:

@techreport{Li:CSD-94-852, Author = {Li, Ren-Cang}, Title = {New Perturbation Bounds for the Unitary Polar Factor}, Institution = {EECS Department, University of California, Berkeley}, Year = {1994}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html}, Number = {UCB/CSD-94-852}, Abstract = {Let <i>A</i> be an <i>m</i> x <i>n</i> (<i>m</i> >= <i>n</i>) complex matrix. It is known that there is a unique polar decomposition <i>A</i> = <i>QH</i>, where <i>Q</i>*<i>Q</i> = <i>I</i>, the <i>n</i> x <i>n</i> identity matrix, and <i>H</i> is positive definite, provided <i>A</i> has full column rank. This note addresses the following question: how much may <i>Q</i> change if <i>A</i> is perturbed? For the square case <i>m</i> = <i>n</i> our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's (<i>SIAM J. Matrix Anal. Appl., <b>14</b> (1993), 588-597.</i>). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.} }

EndNote citation:

%0 Report %A Li, Ren-Cang %T New Perturbation Bounds for the Unitary Polar Factor %I EECS Department, University of California, Berkeley %D 1994 %@ UCB/CSD-94-852 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html %F Li:CSD-94-852